Computing Variational Bounds for Flow through Random Aggregates of Spheres
Abstract
Known formulas for variational bounds on Darcy's constant for slow flow through porous media depend on twopoint and threepoint spatial correlation functions. Certain bounds due to Prager and Doi depending only a twopoint correlation functions have been calculated for the first time for random aggregates of spheres with packing fractions ( η) up to η = 0.64. Three radial distribution functions for hard spheres were tested for η up to 0.49: (1) the uniform distribution or "wellstirred approximation," (2) the PercusYevick approximation, and (3) the semiempirical distribution of Verlet and Weis. The empirical radial distribution functions of Bennett and Finney were used for packing fractions near the randomclosepacking limit ( η_{RCP} ∼ 0.64). An accurate multidimensional Monte Carlo integration method (VEGAS) developed by Lepage was used to compute the required twopoint correlation functions. The results show that Doi's bounds are preferred for η < 0.10 while Prager's bounds are preferred for η > 0.10. The "upper bounds" computed using the wellstirred approximation actually become negative (which is physically impossible) as η increases, indicating the very limited value of this approximation. The other two choices of radial distribution function give reasonable results for η up to 0.49. However, these bounds do not decrease with η as fast as expected for large η. It is concluded that variational bounds dependent on threepoint correlation functions are required to obtain more accurate bounds on Darcy's constant for large η.
 Publication:

Journal of Computational Physics
 Pub Date:
 October 1983
 DOI:
 10.1016/00219991(83)900219
 Bibcode:
 1983JCoPh..52..142B